Notes on congruences on regular semigroups.I
Robert J. Koch, Bernard L. Madison · Journal of the Australian Mathematical Society Series A Pure Mathematics and Statistics · 1985
Abstract Four properties of congruences on a regular semigroup S are studied and compared. Let R, L and D denote Green's relations and let V = {(a, b) ∈ S × S|a and b are mutually inverse}. A congruence ρ on S is (1) rectangular provided ρ ∩ D = (ρ ∩ L) ° (ρ ∩ R), (2) V-commuting provided ρ ° V = V ° ρ, (3) (L, R)-commuting provided L ° ρ = ρ ° L, and R ° ρ = ρ ° R, and (4) idempotent-regular provided each idempotent ρ-class is a regular subsemigroup of S. A rectangular congruence is (L, R)-commuting and a V-commuting congruence is idempotent-regular. If ρ is idempotent-regular and (L, R)-commuting then ρ is V-commuting. Examples and conditions are given to show what other implications among the four properties hold. In addition to characterizations of the properties, these are studied in the presence of other conditions on S. For example, if S is a stable regular semigroup, then each congruence under D is rectangular.